Proof of "Every Cauchy Sequence is Bounded

[fderingoz]

I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from

http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded

I don't understand that how we can take m=N$_{1}$ while m>N$_{1}$ ?

In fact i mean that in a metric space (A,d) can we say that

[$\forall$m,n>N$_{1}$$\Rightarrow$ d(x$_{n}$,x$_{m}$)<1]$\Rightarrow$[$\forall$n$\geq$N$_{1}$$\Rightarrow$ d(x$_{n}$,x_{$_{N_{1}}$})<1]

 

[Erland]

I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from

http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded

I don't understand that how we can take m=N$_{1}$ while m>N$_{1}$ ?

In fact i mean that in a metric space (A,d) can we say that

[$\forall$m,n>N$_{1}$$\Rightarrow$ d(x$_{n}$,x$_{m}$)<1]$\Rightarrow$[$\forall$n$\geq$N$_{1}$$\Rightarrow$ d(x$_{n}$,x_{$_{N_{1}}$})<1]

You are right. This is an error in the wiki. $m,n>N$ should be changed to $m,n\ge N$ wherever it occurs (this also holds with $N_1$ instead of $N$). This fits the wiki's definition of Cauchy sequence, which the wiki's proof doesn't.

 

[fderingoz]

Thank you for the answer i also think like you. This is an error in the wiki. But i saw several functional analysis book which write the proof of proposition same as in wiki. So,

Who is wrong?

 

[Erland]

Well, we can define "Cauchy sequence" with either $>$ or $\ge$, but in the former case, we cannot use $N_1$ the way it is used in the proof in the wiki. Then we also need an $N_2>N_1$ to work with, or something like that.

 

[blue_raver22]

I would first clarify that the statement "every Cauchy sequence in a metric space is bounded" means that the sequence is bounded in the metric space, not just in the set A. This means that there exists a real number M such that for every element x in the sequence, the distance between x and any other element in the sequence is less than M.

In the proof provided on the website, the value N1 is used to establish a bound on the sequence. This means that for any element in the sequence beyond N1, the distance between that element and any other element in the sequence is less than 1. This is necessary to show that the sequence is bounded, as it ensures that the distance between any two elements in the sequence is never greater than 1.

To address the question about taking m=N1 while m>N1, this is simply a notation used to show that m is greater than N1. It is not meant to be taken literally as m=N1.

In response to the second question, yes, in a metric space (A,d), we can say that if for all m and n greater than N1, the distance between x_m and x_n is less than 1, then for all n greater than or equal to N1, the distance between x_n and x_{N1} is also less than 1. This is because for any n greater than or equal to N1, we can always find another element in the sequence, x_{N1}, and the distance between these two elements will still be less than 1.

Overall, the proof provided on the website is a valid and rigorous way to show that every Cauchy sequence in a metric space is bounded. It may be helpful to review the definitions of Cauchy sequences and boundedness in a metric space to better understand the proof.